Module 10.2: nag nlin eqn Roots of a Single Nonlinear Equation. Contents

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1 Nonlinear Equations Module Contents Module 10.2: nag nlin eqn Roots of a Single Nonlinear Equation nag nlin eqn provides a procedure for computing a single root of a continuous function in a given interval. Contents Procedures nag nlin eqn sol Finds a solution of a single nonlinear equation Examples Example 1: Solution of a single transcendental equation Additional Examples References [NP3245/3/pdf] Module 10.2: nag nlin eqn

2 Module Contents Nonlinear Equations Module 10.2: nag nlin eqn [NP3245/3/pdf]

3 Nonlinear Equations nag nlin eqn sol Procedure: nag nlin eqn sol 1 Description nag nlin eqn sol attempts to obtain an approximation to a simple root of the continuous function f(x) given an initial interval [a, b] such that f(a) f(b) 0. This condition implies that there is at least one root of f in the interval [a, b]. The procedure combines the methods of bisection, linear interpolation and linear extrapolation (see Dahlquist and Björck [2]), to find a sequence of subintervals [a i,b i ] of the initial interval such that the final interval contains the root. The approximation, x, totheroot, α, is such that one or both of the following criteria are satisfied: x α < x tol, f(x) < f tol. Since the intervals [a i,b i ] are determined only so that f(a i ) f(b i ) 0, it is possible that the final interval may contain a discontinuity or a pole of f (violating the requirement that f be continuous). This procedure also checks if the sign change is likely to correspond to a pole of f. 2 Usage USE nag nlin eqn CALL nag nlin eqn sol(f, x, a, b [, optional arguments]) 3 A rguments 3.1 Mandatory Arguments f function f must evaluate the function f at the point x. function f(x) real(kind=wp), intent(in) :: x Input: the point x at which the function must be evaluated. real(kind=wp) :: f Result: the value of the function at the point x. x real(kind=wp), intent(out) Output: the approximation to the root. a real(kind=wp), intent(in) b real(kind=wp), intent(in) Input: the lower bound, a, and the upper bound, b, oftheinterval. Constraints: a b, f(a) f(b) 0. [NP3245/3/pdf] Module 10.2: nag nlin eqn

4 nag nlin eqn sol Nonlinear Equations 3.2 Optional Arguments Note. Optional arguments must be supplied by keyword, not by position. The order in which they are described below may differ from the order in which they occur in the argument list. x tol real(kind=wp), intent(in), optional Input: the absolute tolerance to which the root is required (see Section 1). Constraints: x tol > 0.0. Default: x tol = SQRT(EPSILON(1.0 wp)). f tol real(kind=wp), intent(in), optional Input: a value such that if f(x) < f tol, x is accepted as the root (see Section 6.2). Constraints: f tol 0.0. Default: f tol =0.0. error type(nagerror), intent(inout), optional The NAG fl 90 error-handling argument. See the Essential Introduction, or the module document nag error handling (1.2). You are recommended to omit this argument if you are unsure how to use it. If this argument is supplied, it must be initialized by a call to nag set error before this procedure is called. 4 Error Codes Fatal errors (error%level = 3): error%code Description 301 An input argument has an invalid value. 399 An unexpected Library error. Check all procedure calls. If you cannot find any error in your calling program, please report this error to NAG. Failures (error%level = 2): error%code Description 201 f(x) changes sign at a pole. A change in sign of f(x) has been determined as occurring near the point defined by the final value of x. There is some evidence that this sign change corresponds to a pole of f(x). Warnings (error%level = 1): error%code Description 101 Too much accuracy requested. x tol is too small for the computer being used. The final value of x is an accurate approximation to the root. 5 Examples of Usage A complete example of the use of this procedure appears in Example 1 of this module document Module 10.2: nag nlin eqn [NP3245/3/pdf]

5 Nonlinear Equations nag nlin eqn sol 6 Further Comments 6.1 Algorithmic Detail This procedure is a modified version of procedure ZEROIN given in Bus and Dekker [1]. 6.2 Accuracy This depends on the values of x tol and f tol. If full machine accuracy is required, x tol and f tol may be set very small, resulting in an error exit with error%code = 101. This may involve more iterations than are required for a less accurate solution. You should use x tol to control the accuracy (with the default value for f tol) unless you have considerable knowledge of the size of f(x) for values of x near the root. [NP3245/3/pdf] Module 10.2: nag nlin eqn

6 nag nlin eqn sol Nonlinear Equations Module 10.2: nag nlin eqn [NP3245/3/pdf]

7 Nonlinear Equations Example 1 Example 1: Solution of a single transcendental equation To calculate the root of e x x within the interval [0,1]. 1 Program Text Note. The listing of the example program presented below is double precision. Single precision users are referred to Section 5.2 of the Essential Introduction for further information. MODULE nlin_eqn_ex01_mod!.. Implicit None Statement.. IMPLICIT NONE!.. Intrinsic Functions.. INTRINSIC KIND!.. Parameters.. INTEGER, PARAMETER :: wp = KIND(1.0D0) CONTAINS FUNCTION f(x)!.. Implicit None Statement.. IMPLICIT NONE!.. Intrinsic Functions.. INTRINSIC EXP!.. Scalar Arguments.. REAL (wp), INTENT (IN) :: x!.. Function Return Value.. REAL (wp) :: f!.. Executable Statements.. f = EXP(-x) - x END FUNCTION f END MODULE nlin_eqn_ex01_mod PROGRAM nag_nlin_eqn_ex01! Example Program Text for nag_nlin_eqn! NAG fl90, Release 3. NAG Copyright 1997.!.. Use Statements.. USE nag_examples_io, ONLY : nag_std_out USE nag_nlin_eqn, ONLY : nag_nlin_eqn_sol USE nlin_eqn_ex01_mod, ONLY : f, wp!.. Implicit None Statement.. IMPLICIT NONE!.. Local Scalars.. REAL (wp) :: a, b, x!.. Executable Statements.. WRITE (nag_std_out,*) Example Program Results for nag_nlin_eqn_ex01 a = 0.0_wp b = 1.0_wp! Find a zero of the function in the interval [a,b] CALL nag_nlin_eqn_sol(f,x,a,b) [NP3245/3/pdf] Module 10.2: nag nlin eqn

8 Example 1 Nonlinear Equations WRITE (nag_std_out, (/,1X,A,F12.5) ) Zero =, x END PROGRAM nag_nlin_eqn_ex01 2 Program Data None. 3 Program Results Example Program Results for nag_nlin_eqn_ex01 Zero = Module 10.2: nag nlin eqn [NP3245/3/pdf]

9 Nonlinear Equations Additional Examples Additional Examples Not all example programs supplied with NAG fl 90 appear in full in this module document. The following additional examples, associated with this module, are available. nag nlin eqn ex02 Solution of a single transcendental equation, with user-supplied x tol. [NP3245/3/pdf] Module 10.2: nag nlin eqn

10 References Nonlinear Equations References [1] Bus J C P and Dekker T J (1975) Two efficient algorithms with guaranteed convergence for finding a zero of a function ACM Trans. Math. Software [2] Dahlquist G and Björck Å (1974) Numerical Methods Prentice-Hall Module 10.2: nag nlin eqn [NP3245/3/pdf]